Watchman routes under limited visibility

Abstract

In this paper we consider the watchman route problem in simple polygons when there is a distance limit (range) d to the visibility of the watchman. We have then two versions of the watchman route problem. We want to find a shortest route such that either (a) each point in the boundary of the polygon (d- watchman problem) or (b) each point in the polygon (d- sweeper problem) is d-visible (i.e., visible and at most d away) from some point along the route. We first present an O( mn 2) algorithm for the safari route problem which is to find a shortest route that visits a set of m convex polygons that lie in the interior and are attached to the boundary of an enclosing polygon ( n is the total number of vertices). We use this result to obtain a polynomial approximation scheme for the d-watchman problem. The d-sweeper problem is closely related to the traveling salesman problem on simple grids whose complexity is open (it is NP-hard for general grids [8]). We present an approximation algorithm for the TSP in simple grids that obtains solutions within 33% of the optimum. This also provides approximate solutions for the d-sweeper problem.